22 December 2009

**One of a machine's costliest maintenance issues can begin to gel before the equipment ever starts up. How the dynamic support system of a machine is designed will impact the amount of attention a maintenance team must devote to it later on.**

A machine is a machine; the soil on which it rests is unique. Consequently, the support system between a machine and the soil must be designed and engineered to accommodate both the machine and the soil—*and, ultimately, be capable of attenuating vibration*. As a result, many parties, from the OEM to the owner/operator(s) to the maintenance team(s) where machinery runs clearly have a vested interest in assuring that dynamic support systems are specified, designed, fabricated and installed correctly.

Yes! Vibrations can be attenuated. But doing so effectively—*and over the long term*—begins well before the equipment ever starts up. This discussion is intended to provide those in the maintenance and reliability arena with an abridged understanding of structural dynamics and help them see why the design of any successful dynamic support system must be followed by a holistic Finite Element Analysis (FEA) and a Fast Fourier Transform (FFT) of the assembled system.

The FEA is the only way of checking the designed system for its natural frequencies and response to the machine/soil interaction; hence, it allows modifications to be made on paper, not on the prototype in the field. Omitting the FEA increases risk and eliminates natural frequencies and any knowledge of system resonance. Many excuses flourish in the marketplace as to design methods and whether to do or not to do an FEA: "We've always done it that way"... "We work by rule of thumb"... "We cover and execute all the topics required." The question remains: What is required for a better dynamic system? The answer? Including the FEA is mutually beneficial to everyone concerned.

Keep in mind that the natural physics of any dynamic system (regardless of how it is "designed") will lead to natural frequencies based on the system's unique geometry, mass and stiffness distribution. There's always a risk of any of these unknown frequencies being near a driving frequency; whenever they are, resonance problems increase—as does maintenance. Dynamic design and the FEA will circumvent these types of problems.

**Designing**

The duty and function of the design engineer is to predict the operation of a system and to replicate it on paper—*before manufacture*—to confirm satisfactory operation. How the designer accomplishes this is reflected in the following discussion of dynamic support system design and the elements that must be considered.

**Dynamic design guidelines...**

Many guidelines (charts correlating RPM, deflection and machine service) serve two functions.

- First, if a design engineer knows the driving RPM and service desired, appropriate deflections are selected as a design guide.
- Second, after completing the design and FEA, the calculated deflections are compared to the anticipated service.

If deflections are within operational service parameters, the engineer continues with the design. If not within parameters, a redesign is carried out and the FEA is repeated to verify natural frequencies and new service criteria.

The OEM may have such information or provide sources. Displacements are limited by such service data. Example: If at 1000 RPM the designer expects minimum faults, he/she designs to 0.001" of deflection (using the appropriate chart/data as design guide). After the design and FEA, if a resulting deflection is 0.01" at 1000 RPM, the designer would expect faulty to dangerous service (using the appropriate chart/data as an analysis check). This type of procedure will avoid high frequency ratios and amplification factors.

**Design criteria...**

First and foremost, it should be noted that all materials are springs—*steel, concrete, soil*—and must be respected and designed accordingly. Discounting a foundation, deck or slab on grade, etc. (new or existing), claiming rigidity is erroneous. Such an assumption would skew the FEA, resulting in unknown and useless information. It is the duty of the design engineer to control the mass and stiffness of the soil and the complementary foundation.

**Superstructure...**

The superstructure—*steel beams and columns, pedestals, decks, mezzanines, service platforms, motor mounting pads, etc. (new or existing)*—must be dynamically designed. All steel elements for a dynamic design are sized for displacement, a spring effect (not to a stress limit) and must be within selected service conditions of RPM and service quality. Mild steel is sufficient rather than high-strength steel as the modulus of elasticity of all steel is 30x106 psi and deflections are based on the modulus, not strength. Load paths are a very important consideration. Note deflection combinations—*vertically, beams plus columns; horizontally, column plus bracing*—all are plus foundation movement. Bolted connections and/or welds are critical on the load paths, bolts, with slip and/or distortions; welds, use 100% penetration.

Each and every component contributes to the holistic system with its mass, stiffness and location within the structure. All ancillary equipment must be included and symmetrically located to avoid a torsion about a vertical axis in the design and FEA.

**Substructure design...**

A complete substructure (foundation system) includes the founding soil, reinforced concrete pad, pedestals (piers), anchor bolts and grouting systems to receive the structural steel, motors, reducers, etc. Caveat: This rigidity design procedure is vastly different from a standard static foundation design, i.e., 3000 lbs./sq. ft. Be aware of the combined soil/foundation interaction. These two components must be tailored to accommodate the required displacements for RPM and quality of operation. This interaction may become complex to maintain minimal soil displacement and maximum concrete rigidity. The soil rigidity criteria are based on the modulus of subgrade reaction of the soil. It is a measure of soil pressure required for a unit of vertical displacement, having units of lbs./cubic ft., not to be confused with weight, or stated another way,

lbs./sq. ft./ft. (pounds per square foot of load to cause a foot of settlement). The guidance of a geotechnical engineer will be helpful. Modulus of subgrade reaction may range from 30 to 300 k/cubic ft. and must be modified for footing size and type of soil.

The mass and stiffness of all these elements must be included into the dynamic design and FEA of the holistic dynamic system.

**Rigid concrete...**

Rigid concrete slabs, beams, pedestals, etc. require thick (perhaps to 3 ft. or more) concrete sections to maintain deflection limits of 0.001" against movement. This may sound excessive, but remember the modulus of elasticity of concrete is 3x106 psi (one tenth that of steel). Concrete of 3 ft. or more, least dimension, must be designed as mass concrete for internal heat control and expansion. Caveat: Avoid stress risers. A stress riser is a crack failure waiting to happen.

**Analyzing the design****The all-important FEA...**

The FEA is the workhorse of analytical processing required for the solution of the dynamic system. Prior to the advent of the computer, dynamic analysis was impossible—*since that time, our thinking and understanding of dynamic systems has changed greatly*. Unlimited numbers of elements, materials and components are now a reality in structural dynamics.

**FEA INPUT**encompasses all system geometry in the three dimensions (X, Y, Z globally and locally), member sizes, shapes, properties and materials. Steel input is a rather straightforward material to work with; concrete and soil, however, are more complex to design and input into FEA.**FEA OUTPUT**Natural frequencies (eigenvalues), their corresponding shapes (eigenvectors), displacements, translations and rotations are products of the FEA. Natural frequencies are listed in ascending numerical order from the lowest. This is a critical requirement in that natural frequencies near any driving frequency risk the resonance problem. To avoid resonance, the lowest natural frequency should be approximately three times (3x) the highest driving frequency.**FFT**Fast Fourier Transform requires all driving frequencies be loaded and activated in the FEA. FFT will present near resonance spikes in a frequency domain, at their approximate frequency and magnitude (see Fig. 1). These spikes require evaluation for resonance proximity to their FRs and AFs.

**Fig. 1. FFT Plot of resonance spikes at various frequencies. Note the largest at approximately 3.00 Hz.**

**FREQUENCY RATIO (FR)**The frequency ratios require the natural frequencies from the FEA; consequently, the importance of the analysis. The FR = driving frequency fd / natural frequency fn. The driving frequency may be comprised of multiple RPMs, plus beating between various frequencies. There are as many FRs as there are natural frequencies in the system; not all are needed. Caveat: The first (lowest) frequency ratio must be less than 0.33 for all frequencies to be attenuated. FRs larger than 0.33 will be troublesome in the resonance range—*ranging anywhere from problematic to dangerous*. To accomplish the 0.33 FR, the lowest natural frequency of the system must be three times higher than the highest driving frequency. Could that be why dynamic designs are often done using a factor of about three times the weight of the machinery load? Be careful how you think of this. The threes are coincidental —*3x the machinery load (used in some designs) has no consistent known relevance to the proposed dynamic design problem at hand*. The 3x driving frequency proposed in this discussion is valid. Caveat: The finite element analysis must not be omitted with a dynamic design as it often is with a static design. DO NOT confuse or interchange the definition or use of the 3 factor.**AMPLIFICATION FACTOR (AF)**The AF is a multiplier derived from the frequency ratio and applied to the normal static load. As the frequency ratio approaches 1, the AF will spike at resonance accompanied by problems and continuous maintenance (see Fig. 2). Remember this: the nearer to resonance, the nearer to disaster. Example: A bearing, or any other component within the system, is specified to sustain a normal load while a vibration manifests itself near the bearing. The bearing fails, is replaced, fails again, is replaced again and again. The problem is not with the bearing, but rather the structure. If at that bearing location the AF is approximately 7.5, the bearing is loaded at 7.5 times its static load. Why did it fail? It is undersized. Does the bearing OEM understand that? An FEA and AF with a proper dynamic design would have predicted and prevented that problem. Would someone buy a two-ton truck to carry a 15-ton load safely for an extended time? He/she could simply plan for a 15-ton load, buy a bigger truck or slow down the system and make 7.5 trips rather than one. In your case, you might slow the RPM on the machine in question to avert a problem, but to what RPM? Only the FEA and the AF hold that secret—*it is attenuation.*

**Attenuate (v)** — to lessen the amount, or force, of...to reduce the severity of... **Attenuator (n)** — a device for attenuating...reducing the amplitude of...

A proper dynamic design will attenuate the amplitude of vibrations. The subtle device attenuating vibrations is a proper holistic dynamic structural engineering design. The trace of the amplification factor is shown in Fig. 2.

**FR < 1 AF = 1/ (1-FR2)(System will attenuate vibrations.)**

**FR > 1 AF = 1/ (FR2-1)(Part of the system will resonate.)**

**FR = 1 AF will be at resonance and destruction.**

If FR = 0.33 (3x faster than operating RPM) the AF = 1.12 or 12% larger than the static load, maintenance and downtime expense is minimized. The resonance zone, FR of 0.33 or higher, is vulnerable to continuous problematic maintenance—*soft foot, failed anchor bolts, shaft misalignment, steel fatigue, cracked welds, etc.* The FEA will predict the problem allowing a paper redesign. The redesign will require another execution of the FEA process to confirm the adequacy of the modifications.

Another word of caution: Each design change (of a new or existing system) will alter the FEA, change natural frequencies, FRs and AFs, thus requiring a repeat FEA process to eliminate the problems. The resonance spike can only be avoided by the lowest natural frequency being 3x the highest driving frequency. The dynamic design and FEA is a critical part of the design process which is frequently overlooked.

The ODS and MODAL analysis can only be done on the operating prototype in the field and that is a little late to verify a problem you already know by field observation.

**In the end, what does all of this have to do with a maintenance team?**

The maintenance professionals in a plant or facility are the people who will be called on to deal with a machine's vibration problems—*possibly time after time after time*...You and your team might, at some point, also find yourselves being blamed for such a state of affairs. Vibration problems, however, often begin before a machine starts up. Thus, it's up to you, along with the owner/operators and equipment OEMs, among others, to pay attention to the dynamic support system engineering-design process. Are these support systems—*the systems on which your "moneymaker" equipment rests*—being designed by myth, mystery, fact or fiction? Having a good understanding of these systems and how they are specified, designed, fabricated and installed can be invaluable for a maintenance organization. If nothing else, it will allow you and your teams to ask the types of questions that lead to what all interested parties want: good machines that operate better. **MT
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*Paul E. Feuerstein, P.E., is founder and sole proprietor of FEUERSTEIN ENGINEERING in Milwaukee, WI. His practice has focused on vibration, structural dynamics and soil dynamics, among many other unique systems, for 30+ years. Feuerstein has written numerous articles and often presented papers on structural dynamics during his career. He is a Professor Emeritus at Milwaukee School of Engineering, where he served on the faculty for 27 years. For more information, telephone: (414) 964-3034; e-mail
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